The 30-year-old Scot made only one unforced error in winning the opening set and, although he did drop serve late in the second, he broke back immediately to take complete control of the match.
Khachanov had beaten Tomas Berdych and John Isner to make the last 16 but Murray came into the match buoyed by his third-round win over Juan Martin del Potro. "I've come a long way the last 10 days or so". "It's not easy to play on", he added.
Murray converted on his first break point chance and confidently went on to serve the set out to love.
Another break opportunity arrived early in the second set after another couple of rogue forehands and Khachanov ceded the advantage with a double fault.
He felt he stepped up another level against Khachanov, saying: "I think this was probably the best I have played overall". "I sill feel the ankle some times but I am able to run".
Last month, 22 people died when a suicide bomber struck at an Ariana Grande concert in Manchester.
Captain America Congratulates Wonder Woman On Twitter
The only other major release this weekend was Fox's Captain Underpants: The First Epic Movie , which took $23.5m (£18.2m). Before she was Wonder Woman , she was Diana, princess of the Amazons, trained to be an unconquerable warrior.
He said: "It was a bad tragedy in London, also in Manchester".
He proved too experienced for Karen Khachanov, defusing the power of the 21-year-old Russian to win 6-3 6-4 6-4.
South African Anderson, who beat Britain's Kyle Edmund in the third round, was 6-3 3-0 down to Cilic when he pulled out.
Mofils's exit means France's hopes of ending their 34-year-search for a men's champion at Roland Garros will extend into at least 2018.
Wawrinka, the third seed, booked a place in the quarter-finals for the fourth time with a comfortable 7-5, 7-6 (9/7), 6-2 win over Gael Monfils, the last French player in the men's draw.
Saudi, UAE, Bahrain and Egypt cut ties with Gulf state Qatar
Qatar has already assured Japan, one of its largest customers, that the diplomatic feud will not affect shipments of Qatari LNG . Qatar, a small peninsular nation of 2.5 million people, denounced the action as predicated on lies about it supporting militants.
While Murray remembered his loss to Kei Nishikori in the US Open last year, somewhat hilariously Nishikori had no recollection of the encounter, and give that Murray beat him later in the year at the World Tour Finals, there will be an element of revenge.
"I lost my mind, after 6-0 down I knew I had to change something", said Nishikori. I think I played the best match here in Roland Garros against Carla, which always was a tough match. "Sorry that I don't remember much".
The eighth-seeded Nishikori will face the world No. 1 at Roland Garros tomorrow.
Wawrinka took a medical timeout midway through the second set, laying down on the sideline and getting massaged by a trainer, but then played well the rest of the way, using his sweet-swinging one-handed backhand to great effect.
Cilic was leading 6-3, 3-0 after just 48 minutes when Anderson called it quits.
Also making it through was Stan Wawrinka, as the 2016 champion got past home favourite Gael Monfils in straight sets.
Thailand minimally affected by Gulf dispute with Qatar
Reacting swiftly, Pakistan has said that it has no immediate plans to cut its own ties with Qatar and nor should it. In short, Qatar maintains a complicated series of relationships with virtually every single actor in the region.